where \(M_{r_2}\)  is a positive constant that depends only on \(r_2\) . And we do the same for \(\tilde{W}\) . Now, \((m_n,\omega _n,W_n)\)  converges to \((\tilde{m},\tilde{\omega },\tilde{W})\)  in \(\mathcal {C}([0,T], \mathcal {P}_{\delta }(\mathbb {R}^d) ) \times \mathcal {M}([0,T]\times \mathbb {R}^d,\mathbb {R}^d) \times \mathcal {M}([0,T]\times \mathbb {R}^d,\mathbb {S}_d(\mathbb {R})) \) , therefore it also converges in \(\mathcal {M}^+([0,T] \times \mathbb {R}^d) \times \mathcal {M}([0,T]\times \mathbb {R}^d,\mathbb {R}^d) \times \mathcal {M}([0,T]\times \mathbb {R}^d,\mathbb {S}_d(\mathbb {R})) \)  and we can argue as in Theorem 2.34 of [1]} to conclude that \( J_{RP}(\tilde{m},\tilde{\omega },\tilde{W}) \le \liminf _{n} J_{RP}(m_n,\omega _n,W_n)\) . So \((\tilde{m},\tilde{\omega },\tilde{W})\)  is indeed a minimum of \(J_{RP}\)  in \(\mathbb {K}\) .
